HYACINTHOS 26708

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

Denote:

A', B', C' = the midpoints of AP, BP, CP, resp.

(Np), (Na), (Nb), (Nc) = the NPCs of A'B'C', PBC, PCA, PAB, resp.

Ra = the radical axis of (Np), (Na)
Rb = the radical axis of (Np), (Nb)
Rc = the radical axis of (Np), (Nc)

A*B*C* = the triangle bounded by Ra, Rb, Rc

For P = N:

1. ABC, A*B*C* are parallelogic.
Parallelogic centers?
[(ABC, A*B*C*) =: U lies on the circumcircle]

2. ABC, A*B*C* are orthologic.
Orthologic centers?
[(ABC, A*B*C*) = antipode of U in the circumcircle]


[Peter Moses]:

Hi Antreas,

1)
(ABC,  A*B*C*) = X(930).

(A*B*C*,  ABC) = 

a^14 b^2-5 a^12 b^4+11 a^10 b^6-15 a^8 b^8+15 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16+a^14 c^2-10 a^12 b^2 c^2+25 a^10 b^4 c^2-26 a^8 b^6 c^2+7 a^6 b^8 c^2+14 a^4 b^10 c^2-17 a^2 b^12 c^2+6 b^14 c^2-5 a^12 c^4+25 a^10 b^2 c^4-26 a^8 b^4 c^4+5 a^6 b^6 c^4-4 a^4 b^8 c^4+21 a^2 b^10 c^4-16 b^12 c^4+11 a^10 c^6-26 a^8 b^2 c^6+5 a^6 b^4 c^6+2 a^4 b^6 c^6-9 a^2 b^8 c^6+26 b^10 c^6-15 a^8 c^8+7 a^6 b^2 c^8-4 a^4 b^4 c^8-9 a^2 b^6 c^8-30 b^8 c^8+15 a^6 c^10+14 a^4 b^2 c^10+21 a^2 b^4 c^10+26 b^6 c^10-11 a^4 c^12-17 a^2 b^2 c^12-16 b^4 c^12+5 a^2 c^14+6 b^2 c^14-c^16 : : 

 

= lies on these lines: {5,195}

2)
(ABC, A*B*C*) = X(1141).

(A*B*C*,  ABC) =

= a^14 b^2-5 a^12 b^4+11 a^10 b^6-15 a^8 b^8+15 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16+a^14 c^2-10 a^12 b^2 c^2+17 a^10 b^4 c^2-2 a^8 b^6 c^2-17 a^6 b^8 c^2+22 a^4 b^10 c^2-17 a^2 b^12 c^2+6 b^14 c^2-5 a^12 c^4+17 a^10 b^2 c^4-10 a^8 b^4 c^4+5 a^6 b^6 c^4-12 a^4 b^8 c^4+21 a^2 b^10 c^4-16 b^12 c^4+11 a^10 c^6-2 a^8 b^2 c^6+5 a^6 b^4 c^6+2 a^4 b^6 c^6-9 a^2 b^8 c^6+26 b^10 c^6-15 a^8 c^8-17 a^6 b^2 c^8-12 a^4 b^4 c^8-9 a^2 b^6 c^8-30 b^8 c^8+15 a^6 c^10+22 a^4 b^2 c^10+21 a^2 b^4 c^10+26 b^6 c^10-11 a^4 c^12-17 a^2 b^2 c^12-16 b^4 c^12+5 a^2 c^14+6 b^2 c^14-c^16 : : 

 

= lies on these lines: {2,3}, {511,14140}, {5663,16337}, {13391,16336}

Best regards,
Peter Moses